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A NEW REAL - TIME FAULT DETECTION 
METHODOLOGY FOR SYSTEMS UNDER TEST 


PHASE I REPORT 


SUBMITTED BY: 

DR. ROGER W. JOHNSON, Ph.D., PE 
ASSOCIATE PROFESSOR 
MECHANICAL, MATERIALS AND 
AEROSPACE ENGINEERING 
UNIVERSITY OF CENTRAL FLORIDA 
ORLANDO, FL 32816 

SEPTEMBER 19 th , 1998 


CONTRIBUTING GRADUATE STUDENTS: 

SAN JAY JAYARAM 
RICHARD A. HULL 



( 



V 



I 


TABLE OF CONTENTS 


1. SUMMARY 

2. FLIGHT CONTROL 

2.1 Introduction 

2.2 Statement of Work 

2.3 Objectives 

2.4 Approach 

2.5 Flight Control Example 

2.6 System Identification Using MATLAB 

2.7 Error Detection Filter 

2.8 Detection Filter Algorithm 

3. KALMAN FILTERING FOR SIGNAL CONDITIONING 

3.1 Kalman Filter Formulations 

3.2 Study on Optimal Trajectories of Model State Variables 

3.3 Calculation of Optimal Equations 


.1 

.2 

.2 

.2 

.3 

.4 

.4 

.7 

13 

14 

17 

18 
21 
21 


4. SIMULATION RESULTS 


26 



5. LIST OF REFERENCES 


APPENDIX A 

Input - Output Data of Pitch Actuator Controller Test Sequence Used (Figure) 



1.0 SUMMARY 


The subject emphasized herein is application of A New Real - Time Fault Detection 
Methodology for Systems Under Test (SUT) applied to data from the actuator of pitch 
gimbal of a launch vehicle. The technology innovations planned here are software 
intensive and perfectly compatible with the open physical architecture of existing 
monitoring devices and the automated control system of Systems Under Test (SUT). 
Each technology enhancement deals with adding a technical capability without modifying 
existing physical capabilities (unless desired). The essential objective of this research is 
to provide these technical enhancements by handling and evaluating test data in a 
different manner. This equates to operational modifications such as using recursive 
algorithms rather than data smoothing; the use of computer data comparison and 
evaluation techniques instead of monitoring and using human operators for controlling 
mundane test data sequences. To summarize results based on simulations, the error 
detection filter developed is viewed as a viable tool for monitoring and test and 
evaluation environment. The algorithm also includes the capability to monitor and detect 
errors during the steady state response of the system. Some simulation results also shows 
the ability to use this error detection filter algorithm to develop a database of transient 
response modes which could then be used as a comparison with actual system responses 
in real time. 


1 




f 


2 - FLIGHT CONTROL 


2.1 - INTRODUCTION 

The complex automatic systems so widely employed in modern industry can consist of 
hundreds of inter-dependent working parts, which are individually subject to malfunction 
or failure. It is therefore necessary to provide the required operation of the entire system 
by a scheme of monitoring which detects a fault as it occurs, thereby identifying the 
malfunction of the faulty component. The principal concern here is this monitoring 
function, i.e., the detection, prediction and identification of faults during on-line (real- 
time) operation of a dynamic system. 

2.2 STATEMENT OF WORK 

Upon studying and analyzing the previous work done by various engineers on fault 
detection systems, emphasis is given here on a simpler and more effective means of 
automatic fault detection methodology which applies the basic principles of identifying 
the system and using the model based simulations for detecting failures in the system 
components. In the next section, the approach taken is discussed. 

The purpose of this research is focussed on the identification/demonstration of critical 
technology innovations that will be applied to various applications viz. Detection of 
automated machine Health Monitoring (HM), real-time data analysis and control of 
Systems Under Test (SUT). This new innovation using a High Fidelity Dynamic Model- 
based Simulation (HFDMS) approach will be used to implement a real-time monitoring, 
Test and Evaluation (T&E) methodology including the transient behavior of the system 
under test. The unique element of this process control technique is the use of high 
fidelity, computer generated dynamic models to replicate the behavior of actual Systems 
Under Test (SUT). It will provide a dynamic simulation capability that becomes the 
reference truth model, from which comparisons are made with the actual raw/conditioned 
data from the test elements. 


2 



The insertion of this new concept for Health Monitoring (HM) into existing automated 
monitoring and control systems will provide a real-time, intelligent command and control 
system which has the capability to monitor and observe transient behavior along with the 
dynamic parameters of the systems being operated. Current test capability cannot 
measure the dynamic behavior of SUT in real time. Abnormal dynamic properties are 
indicators of an out of tolerance performance of the SUT; they can be a predictor of 
impending failures in those systems. This feature adds a new dimension to existing test 
control mechanizations that will greatly enhance the visibility of the “system state” 
which, in turn, increases the reliability of test and evaluation process over those currently 
in use. This processing technique also promises the real time detection of abnormal data 
flow conditions and the automatic identification of the specific “state” causing the fault 
condition. This attribute will speed up diagnostic analysis to seconds rather than 
minutes/hours, thus reducing significantly, fault detection and diagnosis. 

2.3 OBJECTIVES 

Activities attendant to “industry needs” in the area of intelligent launch command and 
control automated vehicle check-out and system monitoring will be articulated to the 
growing class of commercial launch vehicles. The characterization for the next step in 
evolving the existing launch control processes to a more automated posture is to embed 
these new technical innovations, which makes a high fidelity, dynamic model based 
simulation methodology possible into an automated control system. This effort will 
remove the operator from all mundane process control procedures, and let the computer 
actively control and sequence the subsystem or system under test. It will also provide 
automatic detection of out of tolerance signal flow and furnish the launch operator a 
notion of how close a measurement is out of tolerance in near real time. Taking into 
consideration the state-of-technology, the current status of existing and emerging vehicle 
launch control processes, and the requirements necessary to design an automated, real- 
time methodology compatible to these systems, a set of objectives are identified to satisfy 
these projected needs with selected technology innovations. 


3 



I 


The following objectives will describe the scope of this research: 

1. Identifying the System that is under test and developing a High Fidelity Dynamic 
Model using the input output data sequence of the actual system. 

2. Formulate and apply a new innovation using a High Fidelity Dynamic Model based 
Simulation approach to implement a real time monitoring system for automating a 
detection system for detecting an “abnormal signal flow” in the Engine Flight Control 
System (in particular, the actuator of the pitch axis). 

3. Use a Kalman Filter estimation mechanization to reduce raw measurement error 
before correlation between simulated and actual responses are generated. 

2.4 APPROACH 

The approach taken in the development of detection methodology for automating the 
control system is to employ High Fidelity Dynamic Model Based Simulation (HFDMS) 
method to conduct Test and Evaluation (T&E) procedures. This new innovation of using 
Dynamic models (i.e. those that include the characteristic differential equations along 
with their dynamic parameters) to replicate the behavior of the actual system under test, 
results in a dynamic simulation capability that becomes the reference or truth model, 
from which, comparisons are made with the actual raw data from test elements. If 
detection of an “abnormal flow” is triggered, an automatic hand-over to the designated 
diagnostic component model is implemented for resolution [Ref: 3 and 4], 

2.5 FLIGHT CONTROL EXAMPLE 

A real - time monitoring and error detection algorithm is developed for application with a 
missile control system actuator. A second order discretized model of a first stage rocket 
engine pitch gimbal actuator system is developed. An error detection algorithm and 
filtering approach is then developed which compares the model output to in-coming data 
in real-time. A recursive approximation to the mean square error (MSE) is obtained via a 
discrete low pass filter and used with a dynamic threshold detection algorithm. A novel 
feature of this method is the use of model rate information and a matched filter approach 
to generate the dynamic error threshold. This enables good detection results to be 


4 



obtained for errors in both transient and steady state response characteristics. Actual pre- 
launch data is then used to verify the performance of this error detection filter [Ref: 3]. 


( 


ACTUATOR SUBSYSTEM MODEL: 

The engine pitch controller will be analyzed. The description and block diagram (Laplace 
transform) for this control loop is shown in Fig (1) and (2) respectively. 


AS819D 


AS804C 


POSITIONAL 

COMMAND 


AS216D 


Operational Amplifier 


* 


K 

[ 

w 

— ► 



Low pass filter 



db 

, BE- 



"S 

> 



(ERROR) 
VALVE DRIVE 
CURRENT 


POSITION 

FEEDBACK 


PITCH 

ACTUATOR 




AS819D: Test sequence input. 
AS804C: Servo error signal. 
AS216D: Actuator response. 


Figure (1): Description of engine pitch actuator system. 


5 





{ 



Figure (2): Block diagram of engine pitch actuator system 

The linear closed loop transfer function Y(s)/R(s) for the system in Fig (2) 
[Ref: 1, 2 and 9]: 

m k^col 
R(s) s 2 + 2 £o) n s + a> 2 
where : 



C = —£= 
2jMk]c 


Parameters: 

Y(s) = Laplace transform of the output signal. 
R(s) = Laplace transform of the input signal. 

£ = Damping factor. 

<d„ = Natural frequency (rad/sec) 

k ss = Steady state gain 

M = Mass of actuator/gimbal 

3 = Damping coefficient of actuator 

k a = Actuator gain 

k = Servo Inverter Unit gain 


6 





The engine pitch gimbal actuator system can therefore be modeled as a second order 
system. This model was obtained by using the “ System Identification ” Toolbox for 
MATLAB. The description of the identification methodology is discussed in next section. 
The actual raw data is obtained from a preflight pitch gimbal test sequence shown in 
APPENDIX A. This figure shows the test sequence input commands (AS 819D), the 
actuator response (AS 216D), and the servo error signal (AS 804C). The input command 
sequence consists of a step command, followed by a ramp command that overdrives the 
actuator into its hard limit, then a return to zero command. This sequence is repeated in 
the negative direction. These rates corresponds to sample times of 0.007875 seconds. 

2.6 SYSTEM IDENTIFICATION USING MATLAB 

As stated in the previous section, System Identification toolbox was used to model the 
system under test. The basic steps in system identification: 

a. Collection of raw input-output data sequence from the actual system to be identified. 

b. Selecting and defining a model structure within which a model is to be estimated. 

c. Computation of best model with a given criterion of fit for the known input-output 
data. 

d. Examination of the properties of model thus selected/estimated. 

The control system identification deals primarily with four different types of Model 
structures: 

a. ARX Model structure 

b. ARMAX Model structure 

c. Instrumental variable model structure 

d. Box-Jenkins Model structure. 

In this research, we used the ARX model structure since it was simple and better for 
second order system identifications. Hence a brief discussion is made regarding 
estimating the parameters of an ARX model. 



7 



( 

The parameters of an ARX structure in a general sense are given by: 

A(q)y(J) = B(q)n(t -nk) + e(t) 

Here A(q) is an ny x ny matrix whose entries are polynomials in the delay operator q . It 
can be represented as: 

A(q) = I „ + M~' + + A naq~ na 

Similarly B(q) is an ny x nu matrix 
B(q) = B 0 +B iq -'+ + 

assumption is made that we already know the array z that consists of input/output data 
from the system 
z = [y u] 

For estimating the parameters A(q) and B(q) of the ARX model, use the function arx. 

The command is given by: 
th = arx(z, [na nb nk]) 

Here na, nb and nk are the corresponding orders and delays that define the exact model 
structure. The function arx implements the least squares estimation method using the 
MATLAB for overdetermined linear equations. The input variables y and u are column 
vectors that contain output and input data. The resulting estimated model is contained in 
“ th ” which is called the “theta format ”, This is the basic format for representing models 
in the System Identification Toolbox. It collects information about the model structure 
and the orders, delays, parameters and estimated co-variances of the parameters into a 
matrix. The theta format can further be translated into any other useful model 
representations like Transfer Function representation or State Space representation. For 
more details in this subject, reference can be made to [25 and 14], 


8 



The model representation obtained by using System Identification Toolbox is obtained 
as: 

Transfer function representation: 


C(s) _ 0.81255 + 0.4145 
~R(s)~ s 2 +17.665 + 1.554 

State space representation: 


A=[- 17.66 1 B=[0.8125 C=[l 0] D=0 

-1.554 0] 0.4145] 


Discrete transfer function representation: 


Y(z ) _ 0.0037326z + 0,0035932 
R{z) ~ z 1 -1.8921z + 0.89209 

Figure (3) shows a simple simulation block diagram of the Pitch actuator model with file 
inputs [Ref: 1 1 and 13]. 



Figure (3) : Pitch controller simulation model with input files. 


9 





( 

The parameters found for the nominal system are: 

G = 0.776 

a), = 11.378 rad/sec 

*„ = 0.83 

The Figure (4) represents the data provided, which shows the characteristic response of 
slightly under-damped second order system. This magnified figure also shows that there 
is a digital noise present in the command and response data. 


Pitch Controller Simulation 



Time - seconds 


Figure (4): Pitch actuator controller. 


10 




Figure (5) shows the block diagram of the resulting discretized pitch actuator model. The 
inner block shows the discrete version of the unity gain open loop transfer function, taken 
as the direct Z transform, using sampling time T = 0.007875 seconds of the transfer 
function: 

m < 

R(s) s 2 + 2 <Za> n s 


which yields: 


Y(z) _ 0.0037326 

R{z) ~ z 2 - 1.892 \z + 0.89209 



Gaifll 


Figure (5): Pitch actuator controller model - discrete transfer function (input and output 

limited saturation protection). 


ll 





( 


The output of this transfer function is then limited to a ±3.0 degrees of travel before 
feeding it back to close the loop. The switches shown in the Figure are used to set the 
input to the transfer to zero, whenever the output of the transfer function is being driven 
into the limit. The steady state gain appears as gainl in the diagram and is applied to the 
input command signal. Outputs are provided from the subsystem model to give actuator 
response, actuator rate, and servo error [Ref: 11 and 13]. 

The accuracy of this model is illustrated in Figure (6) and (7), which overlay the input 
command data (from the file), the actual actuator response data (from the file) and the 
actual model response (from the simulation). We can observe that the model is accurate 
enough to demonstrate the methodology of the error detection filter algorithm that is 
discussed in the next section. 


Pitch Controller Simulation 



Time - seconds 


Figure (6): Pitch actuator controller simulation showing model 
response and actuator response (from file). 


12 




Pitch Controller Simulation 



( 


Figure (7): Pitch actuator controller simulation - model response 
and actuator response expanded view. 

2.7 ERROR DETECTION FILTER 

The error detection filter algorithm developed in this study works by comparing the 
actual actuator response data to the response of the actuator subsystem model to the 
actual actuator command data. From the output of the error detection filter, mean squared 
error is computed. This is then compared with the variable threshold which is the “model 
rate of change”. If the mean squared error is less than the threshold value, error detect 
will be “zero”; if the mean squared error value is more than the threshold value, error 
detect will be “one”, which triggers the fault detection system and a fault is detected. 


13 




2.8 DETECTION FILTER ALGORITHM 


( 


A block diagram of the error detection filter process is shown in the Fig (8), which 
represents the simulation diagram for this research study. As shown in this figure, the 
actual command data (obtained either from on line or from a computer file) is used to 
drive the pitch actuator model. The model response and rate of change information are 
input to the error detection filter subsystem, along with the actual response data (obtained 
either on line, or from a computer file). For future simulation purposes, a band limited 
white noise function is shown in summation with the response data. Other blocks in this 
diagram simply represent output ports, which makes the data available for plotting and 
recording. (Figure shown in next page). 



White noise signal 


Figure (8): Pitch controller simulation - file inputs - discrete transfer 

function model 


14 



A detailed block diagram of only the error detection filter is shown in Fig (9). The error 
between the system response data and the model response is squared and passed through 
a discrete low pass filter. This provides and approximation to the expected (or average) 
value of the mean squared error (MSE). This mean squared error value is then compared 
to a threshold value and the error detection filter subsystem outputs a “zero” (meaning no 
error) if the MSE is below the threshold, and a “one” (meaning an error is detected) if the 
MSE is above the threshold [Ref: 1 1 and 13]. 



Figure (9): Error Detection Filter - Recursive mean square - Discrete 
approximation with threshold detection. 


15 

















< 


A novel feature of this detection algorithm is the use of the model rate information to 
dynamically adjust the error detection threshold. When the system is in steady state, it is 
relatively easy to develop a model that accurately represents the system. However, when 
the system is changing state rapidly, it is more difficult to accurately represent the 
system, and more error should be tolerated to prevent false detects. Therefore, as shown 
in the above Figure (9), a dynamic threshold is computed as follows: 


r = r, + 


■V 


where: 

t = Dynamic threshold 

y = Model rate of change 

n = Steady state threshold 

t 2 = Rate sensitive factor 


For this study, the simulation was run for various values of ti and t 2 and a optimum 
values for these constants were achieved. It was found that rj = 0.02 and r 2 - 0.04 
worked well. 

Finally, it should be noted that the dynamic threshold value itself must be passed through 
a discrete digital filter matched to the one used to determine MSE, so that the phase lag 
between the two signals remains matched. This greatly reduces false error detections and 
allows for tighter setting of the threshold parameters. 


16 



( 




( 


3- KALMAN FILTER FOR SIGNAL CONDITIONING 


The use of Kalman filter is uniquely structured to condition raw data sequences in real- 
time. In practice, the following are several cases that can be handled by the recursive 
procedure: 

i. The state variables can be measured directly; the object will be to filter the noise 
from the raw measurement before comparison to the nominal model (false 
detection will be the motivation here). 

ii. The state variables can be measured directly. If several sensors are used to 
measure the same variable, automatic weighing, based on statistical parameters of 
the sensors, is applied by the Kalman filter to generate the “best estimate” of that 
state variable. 

iii. The state variable is impossible to measure directly; this case can be implemented 
by finding the “best estimate” of those states by using measured data that can be 
related by a known functional relationship to the desired state variables. 

The block diagram of the Kalman filter process is shown in Fig (10). 


MEASUREMENT ERRORS 


u 



A PRIORI 
INFORMATION 


Figure (10): Data flow for filtering process. 


17 




3.1 KALMAN FILTER FORMULATIONS [Ref: 5 and 6 ] 

The random process to be estimated is modeled in the form of 

** +1 =fa x t +w k 

The observation or the output of the system is given by, 
z k =H k x k +v k 


where: 

Xk = (nxl) process state vector 
fa = (nxn) state transition matrix 

w k = (nxl) vector, assumed to be white noise with known covariance structure. 

Zk = (mxl) vector measurement (output) 

Hk = (mxn) measurement to state matrix 

v* = (mxl) measurement error assumed to be white noise with known covariance 
structure. 

The covariance matrices for w k and v* vectors are given by Qk and Rk. Assumption is 
made that the initial estimate of the process is known at some point of time and this 
estimate is based on the knowledge of the process. The estimation error is: 

And the associated error covariance matrix is: 

e k = x k - x k 

where - x k - priori - estimate 


With the known apriori estimate, the updated estimate is found: 
P k = E[e k e T k ] = £[(x t - x k )(x k - x k ) T ] 

where: 

Kk = The Kalman filter gain 


18 



i ( 

Now the error covariance matrix with the updated estimate is given by: 

P k = E[e k e T k ] = E[(x t - x k )(x k - x k ) r ] 
x k =x k +K k (z k - H k x k ) 

In terms of noise covariance, Kalman filter gain (K k ) and H k , the error covariance is given 
by: 


P k ={I-K k H k )P-{I-K k H k ) T +K k R k K T k 

Now an expression is found for the computation of the Kalman filter gain ( K k ) by 
differentiating the above equation with respect to K since we wish to minimize the trace 
of P as it is the sum of the mean-square errors of the estimates of all the elements of the 
state vector. Hence K k is given by: 

K k =P k 'H r k (H k P k -H T k +R k y' 

A simple expression can be derived for error covariance matrix from above expressions. 


P k =(I-K k H k )P k - 

The next step is to make the optimal use of the measurement z k+ i. This is accomplished 
by first updating the process estimate: 

= fa*, 

The error covariance matrix associated with updated process estimate is obtained by: 

e k + 1 = **+l — X k* 1 ~^k e k 

Thus we can write the expression for updated error covariance matrix as: 

Pm =APJl +Q k 

The above equations comprises of Kalman filter recursive equations. 


19 



Figure (11) shows the Kalman Filter recursive loop structure. 


Enter prior estimate x 0 and 
its error covariance Pq 





Figure (11): Kalman filter recursive loop. 


Now that we are in a stage where the system is identified using the input output data, a 
model has been developed along with a error detection system and error detection filter 
algorithms, and a stage has been reached wherein we can study the optimal trajectories of 
the model state variables, which will help us in validating the developed model in a 
optimal sense. In other words, we can see from the plots of optimal trajectories with 
varying initial conditions, that the states and the co-state variables of the model reach 
steady state values quite quickly and also all the necessary optimal control conditions are 
satisfied by the state equations. Here study has been made using the state equations and 


20 







( 


forming the Hamiltonian equation to see the behavior of the state and co-state 
trajectories. This is discussed in detail in the next section. 

3.2 STUDY ON OPTIMAL TRAJECTORIES OF MODEL STATE VARIABLES 

The system modeled has representations in State space and Transfer function forms as: 
Transfer Function [Ref: 8]: 

C(s) _ 0.8125s + 0,4145 
R(s)~ s 2 + 17.66s + 1.554 

State Space: 

A=[-17.66 1 B=[0.8125 C=[l 0] D=0 

-1.554 0] 0.4145] 

3.3 CALCULATION OF OPTIMAL EQUATIONS: 

System Equations: 

x, = -17.66x, +x 2 + 0.8125 m 

x 2 =-1.554x, +0.4 145m 

Hamiltonian is assumed to be: 

Hit) = x T ( t)Qx(t ) + M r (t)Ru(t) + p T (t)Apit) + p T it)Bu(t) 
where: 

Q=l, R=1 [assumed] 
p = co-state vector 

The necessary conditions for optimality are: 

dp 

p’m fp'w 

dx 

». 0 

du 


21 



Since the system is of order 2x2, the number of optimal equations we get are four, which 
are: 

£,* =-17.66/7,* +2*,* 
p\ = -1 .554/7* +2xj 

— = 0 = 2u * (0 + 0. 8 1 25 pi (/) + 0.4 1 45/7 2 * (t) 
du 

simplifying => 

u\t) = -0.40625 pi (/) - 0.20725/?; (/) 

Substituting this optimal control in state equations, we get: 

*;(/) = -17.66*; (o + *;</) - 0.33/?; <o - oampXo 

x\(t) = -1.554*; (0-0. 1684/7,* (0-0.0859^(0 

Hence the four optimal differential equations obtained are: 

*,*(/) = -17.66*; (0 + x 2 *(0 - 0.33/7,* (0 - 0.1684/7^(0 
* 2 *(0 = -1.554*; (0-0. 1684/7,* (O-0.0859/7 2 *(O 
pi = -17.66/7,* +2*,* 
p\ = -1.554 p\ +2* 2 * 


By solving the above differential equations by the method of finding eigenvalues and 
eigenvectors, the resulting equations are in the form: 

p\ 

P\ 

♦ 

* 

L*2 

Where the x's shown in the above equations are the eigenvectors of the differential 
equations and X ’s are the corresponding eigenvalues. 

A code is written in Matlab to generate various plots for the optimal states, optimal co- 
states and optimal control trajectories for different initial conditions. The program is 
shown below, by which for different initial conditions, various values of the constants 
can be found out: 


= [*> ]** +c 2 [* 2 ] e*+c 3 [* 3 > v + c 4 [* 4 ]e v 


22 



( 


The matrix of eigenvectors and eigenvalues: 


Eigenvectors : 


V = 


-0.3153+ 0 . 1977i 
-0.0246+ 0 . 024 li 
0.4474+ 0 . 8125i 
0.0032- 0 . 0028i 


-0.3153- 

-0.0246- 

0.4474- 

0.0032+ 


0.19771 
0 . 024 li 
0 . 8125i 
0 . 0028i 


-0.0242 

-0.5622 

-0.0028 

-0.8267 


-0.0072 

0.0513 

-0.0009 

0.9987 


Eigenvalues : 

D = 

0 
0 
0 

1.4512 


-17.6144+ 0 . 8012i 0 0 

0 -17.6144- 0 . 8012i 0 

0 0 -0.1939 

ooo 


Matlab Program: 

%Initial conditions 
a= [ 1 0 0 0] ; 

I=a ■ ; 

v=[-0. 3153+0. 1977i -0. 3153-0. 1977i -0.0242 -0 . 0072 ; -0 . 0246+0 . 0241i - 
0. 0246-0. 0241i -0.5622 0 . 0513; 0 . 4474+0 . 8125i 0 . 4474-0 . 8125i -0.0028 
-0.0009;0. 0032-0. 0028i 0 . 0032+0 . 0028i -0.8267 0.9987); 

V=inv(v) ; 

C=V* I ; 

C1=C (1, : ) ; 

C2=C (2, : ) ; 

C3=C (3, : ) ; 

C4=C ( 4 , : ) ; 

%X1, X2 , PI, P2 are the optimal trajectories. 
t=[0:0. 15:20] ; 

X1=C1* (-0.3153+0. 1977i)*exp( -17. 6144*t+0.8012i*t)+C2* (-0.3153 
- 0 . 1977 i) *exp (-17. 6144*t-0. 8012i*t)+C3* ( -0 . 0242 ) *exp ( 

-0. 1939* t) +C4 * (0.0072) *exp (-1 . 4512*t) ; 

X2=C1* (-0 . 0246+0 . 0241i) *exp (-17 . 6144*t+0 . 8012i*t) +C2* (-0.0246 
- 0 . 0241 i) *exp(-17.6144*t-0.8012i*t)+C3* (-0 . 5622 ) *exp ( 

-0. 1939* t) +C4* (0. 0513) *exp (-1. 4512*t) ; 

P1=C1* (0. 4474+0. 8125i) *exp (-17 . 6144*t+0 . 8012i*t) +C2* (0.4474 
-0.8125i)*exp(-17.6144*t-0.8012i*t)+C3* ( -0 . 0028 ) *exp (-0 . 1939*t ) +C4* ( 
-0.0009) *exp (-1.4512*t) ; 


23 



P2=C1* ( 0.00320. 002 8i ) *exp (17. 6144* t+0. 8012i*t)+C2* (0 . 0032+0 . 002 8i ) *exp{ 
-17.6144*t-0.8012i*t)+C3* (-0 . 8267 ) *exp ( -0 . 1939* t) +C4* ( 0 . 9987 ) *exp ( - 
1.4512+t) ; 

plot (t, XI, ' - ' , t , X2, 1 . , ,t,Pl / , t , P2, 1 * * ) 

title (’Plot of Optimal Tra jectories-State and Costate’) 

xlabel ( 1 time 1 ) 

ylabel (' Optimal Trajectories’) 

gtext ( * XI ' ) 

gtext ( ’ X2 ' ) 

gtext ( ' PI * ) 

gtext ( ' P2 ’ ) 

gtext ('Init Cond: Xl=l, X2=l, Pl^O, P2=0 ’ ) 


Figures (12) and (13) shows plot of optimal trajectories for a set of initial conditions. As 
described before, the above program can be executed for various combinations of initial 
conditions. 


Plot of Optimal Trajectories-State and Costate 



Figure (12): Pitch controller - Optimal trajectories of state and co-state vectors 


24 



Optimal trajectories 


( 


Plot of Optimal Trajectories-State and Costate 



Figure (13) : Optimal Trajectories of State and Co-state vectors with different 

initial conditions. 

Optimal trajectories can be plotted for various initial conditions. Figure (12) and (13) 
shows the trajectories for two such initial conditions. We can observe, by initializing xi 
and others equal to zero, that the optimal state and co-state trajectories reach steady state 
value quite quickly (around 14 seconds). This provides a very good tool to determine the 
behavior and to validate that the model is accurate enough. 


25 



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c 



( 


4 - SIMULATION RESULTS 

The results of a simulation of the nominal system in which no errors were detected is 
shown in Fig (14), and the dynamic threshold nicely brackets the MSE computation 
during the transient response periods. Figure (15) presents the same simulation data, 
showing the behavior of the threshold and MSE during the transient response between - 
and - seconds. Figure (16) is included to show what happens if the dynamic threshold 
value is not filtered to match the MSE, for this same nominal system. This figure 
demonstrates that the time lag due to filtering the MSE can cause a false detect if it is not 
compensated for in computation of the dynamic threshold [Ref: 1 1 and 13]. 

Various cases were studied with varying values of damping factor and the natural 
frequency of the model and the cases are discussed in the next section. 



lime - seconds 


Figure (14): Pitch actuator controller simulation - nominal 
system - no abnormality. 


26 







Time - seconds 


Figure (15): Pitch actuator controller simulation - filtered variable threshold - nominal 
system - no abnormalities - expanded view - (xi = 0.02, x 2 = 0.04) 



Time - seconds 


Figure (16): Pitch controller simulation - No filtering of variable 
threshold -(xl=0.05,x2=0.02) 


27 









It is noted here that the variations were made in the control parameters £ cn, and k SJ to 
test this new innovation threshold idea. These control parameters have been shown to 
relate directly to the physical parameter of the System Under Test, therefore the detection 
of an abnormal control parameter will indicate an abnormal value of the physical 
parameter of the system under test. 

Various figures were discussed to demonstrate the simulation of transient error in the 
system. In one of the demonstrations, different cases have been studied by varying the 
damping factor of the model. Figures show the actuator response, model response and 
error detection flag in each case. During all transient response periods, the error detection 
filter was able to successfully detect and announce the error. Various figures are also 
included to show the dynamic threshold, MSE and the error detect flag in detail for all the 
cases during the transient response periods. 

In this case, variations were made in the damping ratio of the system by considering the 
characteristic equation of the transfer function. Figure (17) shows a case of 75% of the 
nominal damping ratio. Significance notice can be seen, such as increase in the overshoot 
of the model response, which corresponds to an error and we can see that the fault 


Pitch Controller Simulation 



0 5 10 15 2025303540 


Time - seconds 

Figure (17): Pitch actuator controller simulation transient error is set at 75% of nominal 
damping ratio - Detection system signals abnormal response. 


28 




detection is “on”. Figure (18) shows an expanded view of this which clearly shows that 
there is a substantial increase in overshoot; hence the error filtered is triggered to give a 
detection signal. 


3.5 
3 

2.5 


1.5 


I O. 


Pitch Controller Simulation 


■ 

ducad i 

s 

i ratio n 

3 

ons< 

■ 

■ 

■ 


ii 


■ 


■ 

KV 

a 


■ 


■ 


■ 


■ 

m 



■ 


■ 


■ 


■ 

1 



■ 


■ 


■ 


■ 

L 


;tton sif 

mm 

SB 


■ 


■ 


■ 

L 

■ 

■ 

■ 


■ 


■ 




■ 

1 

I 

■ 

■ 

1 

■ 


■ 


1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 

Time - seconds 


Figure (18): Pitch actuator controller simulation - Expanded view of 75% C#. 


Also Figure (19) shows Dynamic threshold vs Mean square error signal (MSE) where 
detection signal is on at the crossing of these two signals. 


Pitch Controiter Simulation 



Figure (19): Pitch actuator controller simulation - Dynamic threshold vs error signal of 


75% Q ,. 


Similar results can be noticed when the natural frequency of the system is varied. In all 
these cases, the error detection filter was able to detect and announce the error 


29 








< ( 

successfully. We can also notice the changes in the model behavior; the transient 
response of the model is oscillatory and settling time is substantially increased (in the 
cases where the natural frequency is less than the nominal natural frequency). In the case 
of Q)> (On, the error was detected to announce the change in the system parameters. 


Overall, the error detection filter algorithm was able to determine the errors when the 
nominal parameters of the system was changed in both the directions (increasing and 
decreasing). 


The basic idea behind all these different cases is the validation of the model and the 
success of the error detection filter to detect and announce error due to the changes in the 
system parameters. By this validation, we come to the conclusion that, if there is any 
abnormal measurement input/output data coming from the actual system under test, the 
model system parameters will change accordingly and due to these changes, the error 
detection filter will detect the faults, hence the basic goal of fault detection is achieved. 


30 



I 


( 



5. LIST OF REFERENCES 

1. B. Friedland, Control System Design, McGrawHill Publications, 1986. 

2. Charles L. Phillips and Royce D. Harbor, Feedback Control Systems, Third Edition, 
Prentice Hall, 1996. 

3. Roger W. Johnson, A New Methodology for Launch/Spacecraft Vehicle Ground 
Health Management, Monitoring and Test and Evaluation (White Paper), Institute 
for Simulation and Training, University of Central Florida, April 29, 1994. 

4. Roger W. Johnson, Sanjay Jayaram, Richard A Hull, A New Real-Time 
Detection/Diagnosis Methodology for Automated Manufacturing, University of 
Central Florida, 1998. 

5. Arthur Gelb, Editor, Applied Optimal Estimation, The M.I.T Press, 1974. 

6. Robert Grover Brown, Patrick Y.C. Hwang, Introduction to Random Signals and 
Applied Kalman Filtering, Third Edition, John Wiley and Sons, Inc., 1997. 

7. Ron Patton, Paul Frank, Robert Clark, Fault Diagnosis in Dynamic Systems - 
Theory and Application, Prentice Hall International Inc., First Edition, 1989. 

8. Donald E. Kirk, Optimal Control Theory - An Introduction, Prentice Hall 
International Inc., First Edition, 1970. 

9. Richard C. Dorf, Robert H. Bishop, Modern Control Systems, Addison Wesley 
Longman Inc., Eighth Edition, 1998. 

10. Bernard Friedland, Maximum Likelihood Failure Detection of Aircraft Flight 
Control Sensors, Journal of Guidance, Control and Dynamics, Vol. 5, No. 5, 1982, 
pp. 498-503. 


31 



( 


11. John N. Little and Alan J. Laub, Control System Toolbox for Use with MATLAB, 
The Math Works Inc., Prentice Hall, 1996. 

12. Lennart Ljung, System Identification Toolbox for Use with MATLAB, The Math 
Works Inc., Prentice Hall, 1997. 

13. The Math Works Inc., SIMULINK - Dynamic System Simulation Software, User’s 
Guide, 1993 and 1997. 

14. The Math Works Inc., MATLAB - High Performance Numeric Computation and 
Visualation Software, 1993. 


32 



( 


APPENDIX - A 



( 


( 


( 


c 



Mr i p 3 n t 



ICOTUU SI 130 86.